Question

You are standing between two speakers that are separated by \(80.0 \mathrm{~m}\). Both speakers are playing a pure tone of \(286 \mathrm{~Hz}\). You begin running directly toward one of the speakers, and you measure a beat frequency of \(10.0 \mathrm{~Hz}\). How fast are you running?

Short answer

Answer: The speed of the person running towards the speaker is approximately 11.66 m/s.

Step-by-step solution

Understanding the concept of beat frequency

Beat frequency is the difference in the frequency heard from the two sources at the listener's position. In this case, the beat frequency measured is 10 Hz, which means the frequency of one source is 10 Hz more or less than the frequency of the other to form a beat pattern. The initial frequency of both speakers, \(f_0\), is given as 286 Hz.

Understanding the Doppler effect for moving observer

The Doppler effect is a phenomenon where the frequency of the sound waves appears to be different for an observer who is moving relative to the source of the sound. The Doppler effect formula for a moving observer can be written as: $$f = f_0 \cdot \frac{v \pm v_0}{v \pm v_s}$$ Where \(f\) is the observed frequency at the listener's position, \(f_0\) is the initial frequency, \(v\) is the speed of sound in medium, \(v_0\) is the speed of the observer (the person running), \(v_s\) is the speed of the sound source (in our case, it is 0 since the speakers are stationary), and the positive or negative sign depends on whether the listener is moving towards or away from the source. Since the person is running towards the speaker, we will use the positive sign.

Find observed frequencies

Since the listener hears a beat frequency of 10 Hz, we can calculate the observed frequencies for both speakers. Let \(f_1\) and \(f_2\) be the observed frequencies of the two speakers. We have: $$f_1 = 286 + 10 = 296 \, Hz$$ and $$f_2 = 286 - 10 = 276 \, Hz$$

Apply the Doppler effect formula

Now we can apply the Doppler effect formula for the observed frequency \(f_1\) towards the speaker the listener is running: $$f_1 = f_0 \cdot \frac{v + v_0}{v}$$ Since \(f_1\) is known, we need to find the speed of the listener, \(v_0\).

Solve for the speed of the listener

To find \(v_0\), we can rearrange the Doppler effect formula: $$v_0 = \left(\frac{f_1}{f_0} - 1\right)v$$ We need to find the speed of sound in the medium, \(v\). The speed of sound in air at room temperature is approximately 343 m/s. We can now plug all the values and solve for the speed of the listener: $$v_0 = \left(\frac{296}{286} - 1\right)343$$ $$v_0 \approx 11.66 \, m/s$$ The speed of the listener (person) running towards the speaker is approximately 11.66 m/s.

Key concepts

Beat Frequency

Imagine you're at a concert with two flutes playing slightly different notes. The fascinating wavering sound that occurs is called a beat. This is a common everyday example of beat frequency, a concept crucial in understanding waves and sound.

In the context of our problem, when two sound waves of close frequencies interfere with each other, they produce beats. These beats are not a separate sound but rather the result of the sound waves' superposition, leading to a fluctuating volume that rises and falls smoothly and repeatedly. The rate of these fluctuations per second is the beat frequency, which you can calculate by taking the absolute difference between the two interfering frequencies.

Let's use our exercise scenario as an illustration: When there’s a difference in the sound frequency coming from the two speakers due to the observer's movement, a beat frequency of 10 Hz is detected. This means that the observer is hearing a rhythmic throbbing of sound 10 times every second, simplifying the analysis of the problem and leading us to the Doppler effect.

Doppler Effect Formula

Ever noticed the pitch of a siren changing as an ambulance drives by? That's the Doppler effect in action, and it's not just for sound—this phenomenon affects all kinds of waves. Moving objects can either rush toward waves, making them seem shorter and higher in frequency, or away, making them seem longer and lower in frequency. This apparent change in wave frequency due to relative motion is what the Doppler effect is all about.

The Doppler effect formula is a mathematical representation of this change. It is given by:
\[\begin{equation}f = f_0 \cdot \frac{v \pm v_0}{v \pm v_s}\end{equation}\]Here, \[\begin{equation}f\end{equation}\] is the observed frequency, \[\begin{equation}f_0\end{equation}\] is the source frequency, \[\begin{equation}v\end{equation}\] is the speed of sound, \[\begin{equation}v_0\end{equation}\] is the observer's speed, and \[\begin{equation}v_s\end{equation}\] is the source's speed. The plus or minus signs account for the direction of the motion. This formula is key to solving problems related to sound waves and motion, enabling us to determine variables like the speed of a moving object based on changes in observed frequencies.

Speed of Sound

Whether you're shouting across a canyon or listening to music, the speed at which sound travels is an essential part of the experience—and it's not always the same. Speed of sound varies with the medium through which it's traveling and environmental factors like temperature and humidity.

In air at room temperature, the speed of sound is approximately 343 meters per second, but if the air is warmer, sound travels faster. In our exercise, the speed of sound is used as a constant value to calculate the speed of the person running toward the speaker using the Doppler effect. By introducing the known speed of sound into the Doppler effect formula, we can solve for the unknown variable: the speed of the observer.

Understanding the speed of sound is crucial for accurately using the Doppler effect formula. Any alteration in the conditions around could lead to a change in the speed of sound, which would, in turn, affect the observer's calculated speed.

Frequency Shift

You’ve likely heard a police siren change pitch as it zooms past. This change is known as frequency shift, and it's a key concept to master when dealing with the Doppler effect and sound waves in general.

A frequency shift is the change in frequency observed when the source of the sound and the observer are in motion relative to each other. In our textbook problem, the running observer experiences a higher frequency when moving toward one of the speakers. This increase in frequency, also known as 'blue shifting', is detectable and measurable. Conversely, moving away from the sound source would cause a 'red shift', where the observed frequency is lower than the source frequency.

Frequency shift not only helps us solve physics problems but also has real-world applications, like in radar and medical imaging technologies. By measuring the shift, we can map out movement and speed, which is exactly what we did to figure out how fast the person is running toward the speaker.